The altitude of a triangle is increasing at a rate of 2500 centimeters/minute while the area of the triangle is increasing at a rate of 3500 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 7000 centimeters and the area is 87000 square centimeters?
x=???
Note: The "altitude" is the "height" of the triangle in the formula "Area=(1/2)*base*height". Draw yourself a general "representative" triangle and label the base one variable and the altitude (height) another variable. Note that to solve this problem you don't need to know how big nor what shape the triangle really is.
so, did you draw a diagram as suggested?
a = 1/2 bh
so, b = 2a/h
da/dt = 1/2 (db/dt * h + b * dh/dt)
Now just plug in your values:
3500 = 1/2 (db/dt * 7000 + 2*87/7 * 2500)
db/dt = -7.88
To solve this problem, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Given that the altitude (height) is increasing at a rate of 2500 centimeters/minute, we can find the rate of change of base by differentiating the area equation with respect to time.
d/dt (Area) = d/dt [(1/2) * base * height]
3500 = (1/2) * base * 2500
Now, we need to solve for the base at the given altitude and area:
87000 = (1/2) * base * 7000
To find the value of base at this point, we can rearrange the equation:
base = (2 * 87000) / 7000
base = 17400 / 7000
base = 2.48 centimeters
Now that we know the value of the base, we can differentiate the area equation with respect to time to find the rate of change of base when the altitude is 7000 centimeters and the area is 87000 square centimeters.
Differentiating the area equation:
d/dt [(1/2) * base * height] = d/dt (Area)
(1/2) * (d/dt(base)) * height + (1/2) * base * (d/dt(height)) = d/dt(Area)
Substituting the given values:
(1/2) * (d/dt(base)) * 7000 + (1/2) * 2.48 * 2500 = 3500
Now we can solve for d/dt(base), which represents the rate of change of the base.
(1/2) * (d/dt(base)) * 7000 = 3500 - (1/2) * 2.48 * 2500
Simplifying,
(1/2) * (d/dt(base)) * 7000 = 3500 - 3100
(1/2) * (d/dt(base)) * 7000 = 400
Finally, we can solve for d/dt(base):
(d/dt(base)) * 7000 = 400 * 2
(d/dt(base)) * 7000 = 800
d/dt(base) = 800 / 7000
d/dt(base) = 0.114 centimeters/minute
Therefore, when the altitude is 7000 centimeters and the area is 87000 square centimeters, the base of the triangle is changing at a rate of approximately 0.114 centimeters/minute.